Introduction to Linear Algebra

Name: Introduction to Linear Algebra
Rating: 4.2 (45 reviews)

From Vectors to Matrices: A Comprehensive Introduction to Linear Algebra

Created byTensor Teach

Last updated 4/2023

English

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What you'll learn

People looking to pursue a career in Data Science
High School or College Students
Aspiring Machine Learning Engineers
Lifelong learners

Course content

4 sections • 13 lectures • 1h 7m total length

Adding & Scaling Vectors5:30
Learn how to add vectors component-wise, perform scalar multiplication, and interpret subtraction as adding a negative multiple, using examples like u = (2,1) and v = (-1,2).
Calculating Vector Lengths & Normalizing Vectors5:48
learn how to compute a vector's length using the pythagorean theorem, and normalize it to a unit vector by dividing by its magnitude.
Dot Products & Matrix Multiplication6:40
Compute the dot product of two vectors via transpose and component-wise multiplication, with a worked example. Then learn matrix multiplication, its output dimensions, and the inner-dimension requirement.
Determinant & Inverse of 2x2 Matrix2:43
Compute the determinant of a two by two matrix by diagonal products minus; then find its inverse by multiplying one over the determinant with swapped diagonals and negated off-diagonals.
Vectors & Matrix Operations

What Is A System of Linear Equations?4:44
Explore what a system of linear equations is through a two-equation example, solve for y, and find the intersection that satisfies both equations.
Solving System of Linear Equations - Gaussian Elimination5:57
Solve a system of linear equations using gaussian elimination on an augmented matrix, perform row operations to a triangular form, then apply back substitution to find x and y.
Cramer's Rule5:21
Apply Cramer's rule to solve a two-by-two system using determinants, replacing columns with constants, and compare with back substitution.
Solving Systems of Linear Equations

Eigenvalues of 2x2 Matrix4:29
Learn to compute eigenvalues of a 2x2 matrix by forming the determinant of lambda times the identity minus A and solving the characteristic equation to obtain eigenvalues 5 and 1.
Eigenvectors of 2x2 Matrix6:53
Compute eigenvectors of a 2x2 matrix by solving (A−λI)V=0, yielding eigenvectors proportional to (−3,1) for λ=1 and (1,1) for λ=5 with any nonzero scalar multiplier.
Diagonalization3:26
Learn to diagonalize a matrix by forming P from eigenvectors and D from eigenvalues. Then compute A = P D P inverse using matrix chain multiplication and verify the decomposition.
Eigenvalues, Eigenvectors, and Diagonalization

What Is A Linear Transformation?5:35
Explore linear transformations as functions that preserve vector addition and scalar multiplication, with rules T(v+w)=T(v)+T(w) and T(c v)=c T(v), including examples from R2 to R1.
Linear Transformations as Matrix Vector Products4:25
Relate a matrix vector product to a linear transformation and connect eigenvalues and eigenvectors through the eigenvalue equation A v = lambda v.
Subspaces, Basis, and Span6:08
Identify subspaces of R2, including the zero vector, and define a basis as a linearly independent generating set whose span is all linear combinations of its vectors.
Linear Transformations & Vector Spaces

Requirements

High School Algebra

Description

Introduction to Linear Algebra is a foundational course designed to provide students with a solid understanding of the fundamental concepts and techniques of linear algebra. Throughout the course, students will explore vectors and matrix operations, systems of linear equations, eigenvalues/vectors, diagonalization, linear transformations, bases, and subspaces, which are all key components of this important mathematical field.

Students will begin by studying the basic properties of vectors and matrices, including how to perform vector addition, scalar multiplication, and matrix operations. They will also learn how to solve systems of linear equations, both algebraically and graphically.

Moving on, students will delve into the concept of eigenvalues and eigenvectors, exploring how they relate to linear transformations and diagonalization. They will also investigate subspaces, bases, and span briefly in order to gain an awareness of the more abstract side of Linear Algebra.

Throughout the course, students will have the opportunity to develop their problem-solving skills through a variety of quiz questions. By the end of the course, students will have a strong foundation in linear algebra that will prepare them for further study in mathematics, engineering, computer science, and other fields that rely on this important subject. No matter whether you are an aspiring data scientist or you are already well into your career, this course can act as both a launching pad and a refresher on key concepts in linear algebra.

Who this course is for:

Beginner Data Scientists

Introduction to Linear Algebra

What you'll learn

Explore related topics

Course content

Vectors & Matrices4 lectures • 21min

Systems of Linear Equations3 lectures • 16min

Eigenvalues & Eigenvectors3 lectures • 15min

Linear Transformations & Vector Spaces3 lectures • 16min

Requirements

Description

Who this course is for: